The classical optimal transport theory studies the 'optimal' way to transport a probability measure to another probability measure with a given cost. Although this has been extensively studied in the classical setting, its free probabilistic analog, where the probability measures are replaced by non-commutative tracial laws, has remained elusive. In this talk, I will explain an analog of the Monge-Kantorowich duality which characterizes optimal couplings of non-commutative tracial laws with respect to Biane and Voiculescu's noncommutative L^2-Wasserstein distance. Then, I will illustrate the subtleties of non-commutative optimal couplings, compared to the classical optimal couplings. Joint work with Gangbo, Jekel and Shlyakhtenko.
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